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Conversation between Alex and Stephen (prev top next)

2007apr15

Alex,

Last week I got a DVD from Ned Resnikoff containing three videos he'd done recently, one of which is going to be shown in a visual music marathon in Boston next weekend (I'm going to fly out to see it a rarity for me; I almost never travel). He read the conversation you and I had been having, and in response to my pictures about frequency ratios, he said "you're drawing your rectangles wrong" and referred me to his take on the rectangle/interval idea. I made the distinction between symbolic/notational and graphical/analogical associations, saying:

There's always some element of the symbolic when you form an association between two things. The question is: how much? The association between words and things, for example, is mostly symbolic, but when you get into onomatopoeia, there are some direct, literal associations ... but it's still a mix ... when you say that "whoosh" is associated with, say, the whooshing of a car going by, the "whoosh" you say has some elements in common with the car's "whoosh" but it's clearly different; you have to understand a symbolic convention: that a sound made by a person is going to map onto real-world, non-speech sounds. So, it's symbolic in that it's a word, but literal in that the word, as a sound, sounds more like the car's whoosh than other words do.

In my bar-graph notation, the symbolic part is that these little rectangles represent notes. The direct/literal part is that time and pitch of the notes are mapped to size and position. Relations between notes like "these go together" and "this pattern is like that pattern" emerge without doing anything extra. By contrast, in one visualization I did, I had a graphic element that meant "this is like that" which I overlaid on the rectangles. That's a symbolic way of doing something that could be done non-symbolically.

Let's say you had a piece of music for two monophonic instruments and you mapped the frequency (in Hz) of the fundamental pitch of one instrument to X and the other to Y, and whenever two notes were sounding, you drew a rectangle with the origin as one corner and the point identified by the two pitches as the opposite corner. Whenever you had a unison, you'd have a square. When you had an octave, you'd have a rectangle with sides 2:1. I'd say this is symbolic in that you're representing pitch by distance, but direct in that equality emerges automatically, without any additional (notational) operations. Or, I should say "near-equality," since 1:1.001 would look about the same. This system doesn't tell you much, since it's hard to estimate the ratios of the sides.

On the other hand, there are mapping rules which display whole-number relations more automatically; Lissajous curves, for example, can show you the identity of fairly complex whole-number relationships. These have other problems for use in musical contexts (since the mechanism by which its forms emerge are not the same as the mechanism by which harmonic sounds sound consonant or dissonant).

He responded by saying that when he was going between aural to visual representations, he tried "to construct an isomorphism between the notation and the phenomena it represents." I agreed that this was a good goal, and asked:

...what phenomena are involved in consonance and dissonance? My take is that the directly perceived parts of this (that is, the parts that aren't learned) are only indirectly related to whole number ratios, and that they are more directly related to beating, critical band, etc. For example, two sine tones in a ratio of 8:17 do not beat, and are only dissonant in the learned sense, but if you add harmonics, then the second harmonic of the 8 (16) beats against the 17 (assuming that the fundamental is in the pitch range where such effects occur). So, my feeling is: to create the right isomorphism, you've got to involve the harmonic structure somehow; otherwise, you risk numerology.

He asked whether "the numerology of whole-number ratios [would be] a good or useful approximation to a model of dissonance?" to which I responded:

Useful, certainly. And, inasmuch as it's useful, good. But beyond that ...

I don't know that I can be an objective judge of it. In a way, it reminds me of my encounter with a new computer user who understood that the icons on the desktop represented files, but didn't understand that it was just a representation --- that the icon was just something the operating system drew to help you understand that there was a file. Understanding that there's a relationship between the icon and something the computer does is useful, and this guy was definitely better off than if he didn't know that there was some connection, but without knowing more about what was really going on "under the hood," he wouldn't be in an optimal position to predict and control his computer's behavior.

Let's take an example. 2:1. The simplest ratio of two different positive integers. Where does using this shorthand work?

Actually, it's easier to map out the swamps than the dry land, so let's ask the opposite question: where does this description fail?

Precision . If one note is at 200 Hz and another is at 102 Hz, does that count as 2:1? It depends. Let's say that the two notes are on a piano, and that the 102 Hz note has a strong 10th harmonic that's about the same amplitude as the 200 Hz note's 5th harmonic. That's going to sound pretty out-of-tune, because those two harmonics will beat twenty times a second. On the other hand, if two people were singing, you probably wouldn't notice (since their pitch would be fluctuating enough to make the beating unnoticeable).

Range . As the pitches involved move outside of a fairly narrow range (and octave or two on either side of middle C), the beating effects change and diminish. An out-of-tune octave between very high or very low pitches won't sound nearly as dissonant as one in the "sweet spot."

Interval . Even if both notes are within a couple of octaves of middle C, the opportunities for harmonics to interact will be fewer the larger the interval is between them. If two pitches are two octaves apart, the first harmonics of the upper note are the 4th, 8th, 12th and 16th harmonics of the lower note, which are typically not as strong as the lower harmonics.

Timbre . For these purposes, timbre means "the amplitude of the harmonics." Since the beating effects are most pronounced when the harmonics involved are close in amplitude, a difference in timbre can have a huge effect on how much beating there is. For the interval of an octave, the frequencies of the harmonics of the two notes are (upper note's harmonics in red ):

1:_, 2:2, 3:_, 4:4, 5:_, 6:6, 7:_, 8:8, 9:_, 10:10...

If the two notes are played by instruments with harmonic amplitudes that fall off as 1/N, the picture is like this:

1:_, 2: 2, 3 :_, 4:4, 5:_, 6: 6, 7:_, 8: 8, 9:_, 10 :10...

The 2:2, 4:4, and 6:6 harmonic pairs both have a chance of beating when the notes are out of tune. The situation is different if you've got a clarinet. It has weak even harmonics, especially the second. So if a clarinet plays one note and another instrument (say, an oboe) plays a note an octave higher, the amplitudes are like this:

1:_, 2: 2, 3:_, 4: 4, 5:_, 6:6, 7:_, 8 :8, 9:_, 10 :10...

The 2:2 pair will only beat significantly if the higher instrument is playing very softly, in which case, its other harmonics won't beat much; if it plays louder, its fundamental will drown out the clarinet's weak second harmonic. Either way, the sense of dissonance will be a lot less than in the previous case.

And, as you suggested, if the notes have harmonics that are not in whole-number frequency ratios with the fundamental (like bells), or if they don't have much harmonic content (like whistling), or if the notes are too short for beating to happen, or the vibrato is wide enough ... the whole thing falls apart.

Of course, there are circumstances where nothing goes wrong, where we have the kind of interaction that would be predicted by the whole-number ratios. But how is it "predicted"? What's the logic of it? Here's a way of saying it (with the wiggle words emphasized):
When two tones have harmonics with frequencies that are close to being whole-number multiples of their fundamental frequencies and the frequencies of their fundamentals are close to being an integer ratio M:N, the Nth harmonic of the first tone (and all its whole-number multiples) and the Mth harmonic of the second tone (and its multiples) will be the close in frequency; if these harmonics have similar amplitudes (and their amplitudes are large enough to have an impact on the sonic whole), we will perceive beating if they differ by an amount in a certain range, and we perceived this beating as dissonance if its rate is within a certain range.
For me, that's a very different explanation than "we recognize notes as consonant because their frequencies are in simple whole-number ratios."

Is the difference between these two descriptions a matter of nuance or a matter of substance? For me, the answer hinges on the fact that there are three sets of whole-number ratios involved (one between the frequencies of the fundamentals of the two notes and one among the harmonics of each note) and the consonance/dissonance effects arise from the interaction of unisons (or intervals near unison) that arise based on the relationships among these sets. If you stop at the whole-number ratios of the fundamentals, you have predictive power, but you've left out the explanation part of the equation.

Here's another system for predicting consonance/dissonance: "the degree of dissonance between notes depends on the absolute mod-12 distance in half-steps between their pitches, increasing in this series: 0, 5, 4, 3, 2, 6, 1." This has roughly the same predictive power as the whole-number-ratio system, and it's simpler for musicians to use, because you don't need to know anything about frequencies. It doesn't give you much insight into why the numbers are what they are, why numbers later on the list are more dissonant ... but I'd argue that the whole-number explanation suffers the same defect.

Would it be sufficient to represent beats? It would certainly be a closer approximation (and I have worked on it a bit and know what I'd try next if I had time) ... but there's still something missing. In this whole discussion, "consonance" is defined as the absence of beating. (For me, this is kind of like saying that virtue is the absence of sin.) Beating can be absent for several reasons, only one of which is that the pitches are in simple whole-number ratios. Most notably: if you play a note on a single instrument, there's no beating. Is consonance merely the situation where it sounds like there's only one instrument? No, there's more to it than that ...

Stephen

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